Computational Study ofSystem Dynamics

Review of Rate LawsJCE Summary

Computational Methods

PDynamics Methods< Chemical Kinetics Simulator< A+BC: General Trajectory

PMathematics Methods< Integral< Differential

PGraphical Interface Differential EquationSolvers

Dynamics MethodsChemical Kinetics Simulator

http://www.almaden.ibm.com/st/msimPCKS does not integrate differential equationsPCKS performs general, rigorously accurate

stochastic algorithm to propagate reactionPSpeed< Comparable in efficiency to integration for simple< Significantly faster for “stiff” systems

PModels complex reactions< Explosions< Changing volumes

PHigh learning curvePResearch quality

Dynamics MethodsA+BC: General Trajectory

http:/qcpe.chem.indiana.edu/cgi-bin/catalog/view.pl

PLEPS diagramPMonte Carlo Collisions

A

B

C

cm

2b

A B A a Bko o⎯ →⎯⎯ = =, 0

− = = −dAdt

kA A ae kt

( )dBdt

kA B a e kt= = − −1

Mathematics MethodsGeneral

P Integral Methods< Use integrated rate laws< Tables and Graphs

PDifferential Methods< Use computational technique to integrate

differential rate laws< Tables and Graphs

Review of Rate LawsSimple First Order Reactions

a = 1000k = 1

t A B0 1000 0

0.1 904.837418 95.1625820.2 818.730753 181.2692470.3 740.818221 259.1817790.4 670.320046 329.6799540.5 606.53066 393.469340.6 548.811636 451.1883640.7 496.585304 503.4146960.8 449.328964 550.6710360.9 406.56966 593.43034

1 367.879441 632.1205591.1 332.871084 667.1289161.2 301.194212 698.8057881.3 272.531793 727.4682071.4 246.596964 753.4030361.5 223.13016 776.869841.6 201.896518 798.1034821.7 182.683524 817.3164761.8 165.298888 834.7011121.9 149.568619 850.431381

2 135.335283 864.6647172.1 122.456428 877.5435722.2 110.803158 889.1968422.3 100.258844 899.7411562.4 90.7179533 909.2820472.5 82.0849986 917.9150012.6 74.2735782 925.7264222.7 67.2055127 932.794487

0 2 4 6 8 10 12t

0

200

400

600

800

1000

1200

AB

a = 1000k = 1

dA=-k A dt A = a - dA B = a - At dA A B

0 0 1000 00.1 -100 900 1000.2 -90 810 1900.3 -81 729 2710.4 -72.9 656.1 343.90.5 -65.61 590.49 409.510.6 -59.049 531.441 468.5590.7 -53.1441 478.2969 521.70310.8 -47.82969 430.46721 569.532790.9 -43.046721 387.420489 612.579511

1 -38.742049 348.67844 651.321561.1 -34.867844 313.810596 686.1894041.2 -31.38106 282.429536 717.5704641.3 -28.242954 254.186583 745.8134171.4 -25.418658 228.767925 771.2320751.5 -22.876792 205.891132 794.1088681.6 -20.589113 185.302019 814.6979811.7 -18.530202 166.771817 833.2281831.8 -16.677182 150.094635 849.9053651.9 -15.009464 135.085172 864.914828

2 -13.508517 121.576655 878.4233452.1 -12.157665 109.418989 890.5810112.2 -10.941899 98.4770902 901.522912.3 -9.847709 88.6293812 911.3706192.4 -8.8629381 79.7664431 920.2335572.5 -7.9766443 71.7897988 928.210201

0 2 4 6 8 10 12t

0

200

400

600

800

1000

1200

AB

a 1000:=

k 1:=

A t( ) a e k− t⋅⋅:= B t( ) a 1 e k− t⋅

−( )⋅:=

0 2 4 6 8 100

500

1000

A t( )

B t( )

t

Mathematics Methods--IntegralSpreadsheet

Mathematics Methods--DifferentialSpreadsheet

Mathematics Methods--IntegralMathcad

k 1:=

Given

tA t( )d

dk− A t( )⋅ A 0( ) 1000

A Odesolve t 10,( ):=

Given

tB t( )d

dk A t( )⋅ B 0( ) 0

B Odesolve t 10,( ):=

0 5 100

500

1000

A t( )

B t( )

t

Mathematics Methods--DifferentialMathcad

Mathematics Methods--IntegralMathematica

Mathematics Methods--DifferentialMathematica

A B C A a B Ck ko o o

1 2 0⎯ →⎯⎯ ⎯ →⎯⎯ = = =,

− = = −dAdt

kA A ae kt

( )dBdt

k A k B Bak

k ke ek t k t= − =

−−− −

1 21

2 1

1 2

dCdt

k B C ak ek k

k ek k

k t k t

= = −−

−−

⎛

⎝⎜

⎞

⎠⎟

− −

22

2 1

1

1 21

1 2

k1 1:= k2 0.5:=

a 1000:=

A t( ) a e k1− t⋅⋅:=

B t( )a k1⋅

k2 k1−e k1− t⋅ e k2− t⋅

−( )⋅:=

C t( ) a 1k2

k2 k1−e k1− t⋅⋅−

k1k1 k2−

e k2− t⋅⋅−⎛⎜

⎝⎞⎠

⋅:=

Note that k1 must be different than k2 in orderfor these integrated equations to work.

0 2 4 6 8 100

200

400

600

800

10001000

0

A t( )

B t( )

C t( )

100 t

Mathematics Methods--IntegralMatlab

Review of Rate LawsConsecutive First Order Reactions

Mathematics Methods--IntegralMathcad

k1 1:= k2 1:=

a 1000:= b 0:= c 0:=

T 10:=

Given

tA t( )d

dk1− A t( )⋅ A 0( ) a

tB t( )d

dk1 A t( )⋅ k2 B t( )⋅− B 0( ) b

tC t( )d

dk2 B t( )⋅ C 0( ) c

NA

NB

NC

⎛⎜⎜⎝

⎞

⎠Odesolve

A

B

C

⎛⎜⎜⎝

⎞

⎠t, T,

⎡⎢⎢⎣

⎤⎥⎥⎦

:=

0 2 4 6 8 100

200

400

600

800

10001 103×

0

NA t( )

NB t( )

NC t( )

100 t

A B C D A a B Ck k k ko o o

1 2 3 4 0⎯ →⎯⎯ ⎯ →⎯⎯ ⎯ →⎯⎯ ⎯ →⎯⎯ = = =K K,

− =dAdt

kA

dBdt

k A k B= −1 2

dCdt

k B k C= −2 3

N c e c e c e c enk t k t k t

nk tn= + + + +− − − −

1 2 31 2 3 K

( )( ) ( )ck k k k a

k k k k k kn

n1

1 2 3 1

2 1 3 1 1

=− − −

−L

L

( )( ) ( )ck k k k a

k k k k k kn

n2

1 2 3 1

1 2 3 2 2

=− − −

−L

L

( )( )( ) ( )ck k k k a

k k k k k k k kn

n3

1 2 3 1

1 3 2 3 4 3 3

=− − − −

−L

L

Mathematics Methods--DifferentialMathcad

Review of Rate LawsSeveral Consecutive First Order Reactions

!

Bateman solution:

!

Graphical Interface DifferentialEquation Solvers

General

PModel differential rate lawPSoftware solves differential equations< Watch “stiff”

PTables and Graphs

Graphical Interface DifferentialEquation Solvers

Stella

Graphical Interface DifferentialEquation Solvers

Berkeley Madonna

Graphical Interface DifferentialEquation Solvers

VisSim (Mathcad)

Graphical Interface DifferentialEquation Solvers

Simulink (Matlab)

A B C A a B b Cko o o+ ⎯ →⎯⎯ = = =, , 0

− = − = =dAdt

dBdt

kABdCdt

kAB

Graphical Interface DifferentialEquation Solvers

Simile

Review of Rate LawsSimple Second Order Reactions

Graphical Interface DifferentialEquation Solvers

Change of Paradigm

Introduction toSTELLA

Structural Thinking Experiential LearningLaboratory with Animation

Commerical ProductsGraphical Interface Differential Equation Solvers (GIDES)

PStella< Free run-time version

PBerkeley Madonna< Free run-time version< Solves “stiff” differential equations

PVisSim (Mathcad)PSimulink (Matlab)PSimilePModel Maker

Stella InterfaceStella “Layers” – Modeling

PConstructModel usingBuilding Blocks,Tools, Objects

POutputs

Layer Navigation

Map/Model ToggleChange to P2 (or else!)

Run Controller

Stella InterfaceStella “Layers” – Mapping

PTextPOutput Tables

and GraphsP Input using

Slides andDials

PPicturesPQuick Time

Movies

Stella InterfaceStella “Layers” – Equation

P Initial Values ofStocks

PConnectorInformation

PDifferentialEquationsRepresentingthe TimeDependence ofStocks

Stella InterfaceMenu / Icons

Stella InterfaceBuilding Blocks – Stocks

Value Undergoing a Change

Stella InterfaceBuilding Blocks – Flow

Change of Stock with Respect to Time

Stella InterfaceBuilding Blocks – Converter

ConstantsTransformation Equations

Stella InterfaceBuilding Blocks – Connectors

Links Building Blocks

Stella InterfaceTools – Hand

General Purpose Editing ToolCursor Pointer

Stella InterfaceTools – Dynamite

Delete (No “Undo”)

Stella InterfaceObjects – Graph Pad

Graphing Results

Stella InterfaceObjects – Table Pad

Spreadsheet-like Table of Results

Solving of Differential EquationsTaylor Series

Solving of Differential EquationsEuler Method

First 2 terms

Solving of Differential EquationsRunge-Kutta 2

First 3 Terms

Solving of Differential EquationsRunge-Kutta 4

First 5 Terms

Simple Model – Falling CalculatorSystem

y = heightyN = velocity = (acceleration)(time)yO = acceleration = g = 9.8 m s-2

Simple Model – Falling CalculatorStella Model

Simple Model – Falling CalculatorStella Model – Equation Layer

dCdt

kC

C Ckt

o

= −

= +

2

1 1

Simple Model – Falling CalculatorStella Model – Accuracy

PAir FrictionPBounce

Chemical KineticsSure looked like a natural fit to me!!

Kinetics ModelDimerization of Cyclopentadiene2nd Order Diels-Alder Reaction

2

Kinetics Model

Michaelis-MentonLen Soltzberg (Simmons College)

E + S º (ES)

(ES) 6 P + E

Oscillating ReactionsCriteria

PTwo or more coupled reactionsPAutocatalytic

Oscillating ReactionsBriggs-Rauscher

PH2O2

PKIO3 and H2SO4

PHOOCCH2COOH, MnSO4, starch

~15 s for each cycle

Oscillating ReactionsBelousov-Zhabotinskii

P 18 reversible stepsP 21 different chemical species

BrO3- + HBrO2 + H3O+ 6 2 BrO2 + 2 H2O

2 BrO2 + 2 Ce3+ + 2 H3O+ 6 2 HBrO2 + 2 Ce4+ + 2 H2O

A + Y 6 X + PX + Y 6 2PA + X 6 2X + 2Z2X 6 A + PB + Z 6 (f/2) Y

whereA = BrO3

-

X = HBrO2Z = Ce4+

P = HOBrB = organicY = Br -

Oregonator by Len Soltzberg (Simmons College)

Other ApplicationsRod Schluter (Formerly of CofC)

Acid/Base Equilibrium

H+ + OH- 6 H2O

D Tu

xe

dxD x( / )θ =−∫

313

3

( )

( )[ ]

U U RTD

C R Du

e

S R D e

A U RT e D

v u

u

u

− =

= −−

⎛⎝⎜

⎞⎠⎟

= − −⎡⎣⎢

⎤⎦⎥

− = − −

−

−

0

0

3

3 4 31

343

1

3 1

ln

ln

Other ApplicationsDebye Theory for Monatomic Crystals